## Description

• The lab must be completed in R Markdown. Create a pdf file (via ’knit to pdf’) and use

it as your lab submission.

We have seen both autoregressive (AR) and moving average (MA) models, as well as hybrids

of these known as autoregressive moving average (ARMA) models. While these are quite versatile

models, recall that they are used to model series that are stationary.

1. Without using any mathematical notation, describe in words what it means for a time

series to be stationary.

2. Consider a realization of a process with values given in file lab4data.csv. Read in the

data, coerce it into a ts object, plot and comment on whether the series appears to satisfy

the requirements of stationarity.

3. One common way of removing a trend is to difference the data, where instead of looking

at the time series {xt} we look at {yt} with

yt = ∇xt = xt − xt−1.

(Note this series will have one less term than the original.) Using the data above, determine

and plot the differenced time series {yt}. Comment on the resulting plot and the acf of {yt}.

A useful function here is diff(x, lag=1, difference=1), which returns suitably lagged

and iterated differences. Use R help to learn about options lag and differences.

4. In order to remove a seasonal effect, we could difference over the seasonal period. For

instance, take our de-trended data {yt} and difference again but at a lag equal to the

seasonal period, s. The new series will be

∇syt = yt − yt−s,

where s is the period of the seasonal effect. Note the series will again become shorter,

this time with s fewer terms. Choosing the appropriate value of s here, apply seasonal

differencing to the series from part 3 and plot the resulting series. Plot the acf of ∇syt

.

Does your ∇syt resemble white noise?

5. Suggest which type of model from the SARIMA family you would use for the original data.

6. (Theoretical exercise) We have seen that removing trends and seasonality can be as

simple as differencing successively at different lags. In such cases, suitable models are

integrated ARMA (ARIMA) models, or SARIMA if we include seasonal differencing. One

difficult aspect of these models is describing them in mathematical terms. Let us try

re-using the notation we used with ARMA models to describe these processes.

(a) Firstly, we took our original time series Xt and differenced it to obtain Yt = Xt−Xt−1,

then differenced again at a longer lag to obtain Wt = Yt − Yt−s. Combine these two

operations to express our final model Wt

in terms of the original series Xt

.

(b) Recall the differencing operator B which takes a series and returns the series at lag 1.

For instance, BXt = Xt−1. Express the differenced series Yt

in terms of B and Xt

.

(c) Now if BsYt = Yt−s, express your answer from (a) in terms of B. That is, write Wt

in

terms of B, s, and Xt

.